Events This Week

Applied and Computational Mathematics

Improving Model Selection Using Geometric Data Characterization

Brandilyn Stigler - Southern Methodist University

Abstract:

Inferring models from experimental data is an important problem in systems biology. While there are many classes of functions that can fit data from an underlying network, under certain conditions all functions fitting discretized data are polynomials. A useful consequence is that polynomial models of discretized data can be written in terms of a monomial basis, where each choice of basis provides a different prediction regarding network structure.

In this work, we use affine transformations to partition data sets into equivalences classes with the same sets of monomial bases. This partition reveals a geometric property of data sets that have unique models associated with them. Implications of this work are guidelines for designing experiments which maximize information content and for determining data sets which yield unambiguous predictions.

Location: Gibson Hall 310

Time: 3:30

Thursday, December 6

Colloquium

The power of statistics in genetics: examples from forensics, precision medicine, and trans-ethnic studies

The last two decades have brought swift change in the area of human genetics. For less than $100 per person, genotypes for over 2 million genetic markers can be obtained, and with large publicly available genetic reference panels, a person's genetic data can be further imputed to well over 7 million genetic markers. This data revolution has opened up many scientific and statistical questions, though deciphering such a large amount of information can be daunting and requires an understanding of both genetics and statistics. Here, I provide motivation and context to the scientific question and detail the statistical methods and results in three areas: the use of familial matching in forensics, discovery of disease sub-types, and the combination of genetic findings over studies of individuals with differing ancestry. Finally, I end with some open research questions in statistical genetics, with a particular motivation towards translating the wealth of genetic data and results into public health and medicine.

Location: Gibson Hall 325

Time: 3:30

Wednesday, December 5

Algebra and Combinatorics

On the regularity of binomial edge ideals of graphs. Part 2

A.V. Jayanthan - Indian Institute of Technology, Madras

Abstract: TBA

Location: Gibson Hall 325

Time: 3:00

Wednesday, December 5

Probability and Statistics

The tempered fractional Lévy process

Cooper Boniece - Tulane University

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The celebrated family of fractional stochastic processes exhibits a phenomenon called long-range dependence, or slow decay of their covariance as the time separation between two observations increases.

In recent development have been tempered fractional models, which exhibit semi-long range dependence — i.e. their covariance exhibits slow decay over an intermediate time scale, but ultimately transitions to exponential decay. Tempered models have recently found applications in nanobiophysics and finance, where this type of transient behavior is observed in real data.

In this talk, I will introduce a new stochastic process called the tempered fractional Lévy process (TFLP) and discuss some of its probabilistic properties. This is joint work with Farzad Sabzikar (Iowa State) and Gustavo Didier (Tulane).

Location: Gibson Hall 126

Time: 3:00

Tuesday, December 4

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Monday, December 3

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Week of November 30 - November 26

Friday, November 30

Special Colloquium

Moving through a turbulent environment: Embedding models in real-world data

Mimi Koehl - UC Berkeley (host:Fauci, Lisa)

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When organisms locomote and interact in nature, they must navigate through complex habitats that vary on many spatial scales, and they are buffeted by turbulent wind or water currents and waves that also vary on a range of spatial and temporal scales. We have been using the microscopic larvae of bottom-dwelling marine animals to study how the interaction between the swimming by a microorganism and the turbulent water flow around them determines how they move through the environment. Many bottom-dwelling marine animals release tiny larvae that are dispersed to new sites by ambient water currents. To recruit to new sites on the sea floor, these larvae must leave the water column and land on surfaces in suitable habitats. We are studying the mechanisms larvae use to move through turbulent flow and land on surfaces. Field and laboratory measurements enabled us to quantify the fine-scale, rapidly-changing patterns of water velocity vectors and of chemical cue concentrations near coral reefs and along fouling communities (organisms growing on docks and ships). We also measured the locomotory performance of larvae of reef-dwelling and fouling community animals, and their responses to chemical and mechanical cues. We used these data to design agent-based models of larval behavior. By putting model larvae into our real-world flow and chemical data, which varied on spatial and temporal scales experienced by microscopic larvae, we could explore how different responses by larvae affected their transport into reefs or fouling communities. The most effective strategy for recruitment depends on habitat.

Location: Gibson Hall 310

Time: 3:30

Thursday, November 29

AMS/AWM Faculty Talk

Explorations in elastohydrodynamics at the microscale

Lisa Fauci - Tulane University

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The motion of flexible filaments in a fluid environment is a common element in many biological systems. Examples include diatom chains moving in the ocean, bacterial flagella propelling a cell body, cilia in pulmonary airways, and embedded polymers in a complex fluid. In cases where these flexible filaments move through confined environments, the confinement could have a dramatic effect upon the dynamics of the system. We will present recent results on the computational modeling of such systems. Location: Gibson Hall 400-D

Time: 1:30

Thursday, November 29

Applied and Computational

Cell migration from birth to death: Modeling and analyzing the motion of cells in tissues and tumorsc

Tracy Stepien - University of Arizona

Abstract:

Beginning momentarily after we are conceived through to our final days, cells migrate within our bodies. From embryonic development to the progression of many diseases including cancer, cell migration plays an essential role in maintaining our health. To understand the mechanisms and forces involved in migration related to early embryonic development, eye and retina development, wound healing, and cancer growth, I have developed continuum mechanical models with free boundaries and reaction-diffusion equation models of the spread of tissues and tumors. Mathematical analysis and numerical simulations of the models indicate conditions for traveling wave and similarity under scaling solutions, and data and image analysis of experimental data has facilitated the estimation of model parameter values that are physically relevant. In this talk, I will give examples of biological cell migration problems that I work on as well as an in-depth look at some of the mathematical analysis that has arisen from the model equations.

Location: Gibson Hall 325

Time: 2:30

Thursday, November 29

Special Applied and Computational

Asymptotic approximations of near fields in scattering problems

Camille Carvalho - UC-Merced

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Accurate evaluations of near fields can is crucial in a wide range of applications, like for the modeling of micro-organisms swimming in Stokes flow, or for the light enhancement in plasmonic structures. Plasmonic structures are in particular made of dielectrics, and metals (or metamaterials) for which the electromagnetic properties enable the propgation of highly-oscillating (sub-wavelength) surface waves at the interface of the two materials. Boundary integral equation methods can approximate the solution of such problems with high-accuarcy using Nystöm methods, however this accuracy is lost for evaluation points close (but not on) the boundary. In this presentation we present some technique based on asymptotic approximations to address the close evaluation problem for acoustic scattering (Helmholtz), and we will discuss the case of scattering in plasmonic structures at the end.

Location: Stanley Thomas 316

Time: 11:00

Wednesday, November 28

Special ColloquiumSpecial

This talk presents two Bayesian approaches to making efficient statistical inferences for genome-wide association studies (GWAS) using regression models. The first method concerns the marginal analysis of each covariate, while the second is for the joint analysis of the entire dataset.

In the first work, motivated by the "Bayes/non-Bayes compromise", we characterize the asymptotic distribution of the Bayes factor in linear regression with conjugate priors. We show that, under the null, the log(Bayes factor) is distributed as a weighted sum of independent chi-squared random variables with a shifted mean. This enables us to analytically evaluate the p-value associated with a Bayes factor. By implementing a recent algorithm of Bausch (2013), we are able to compute extremely small p-values to arbitrary precision. Moreover, our result helps explain Bartlett’s paradox and the prior-dependence nature of the Bayes factor.

The second work concerns Bayesian variable selection for linear regression models using Markov chain Monte Carlo methods (MCMC). The key innovation is a novel iterative algorithm for solving a special type of linear systems, which is the most time-consuming step in each MCMC iteration. We call our method iterative complex factorization (ICF) and prove that ICF always converges. In the context of Bayesian variable selection with MCMC, ICF is much faster and more accurate than other iterative methods such as Gauss-Seidel iterations. Our algorithm is particularly useful for the heritability estimation with massive GWAS datasets, which can be prohibitively slow via traditional MCMC methods.

Location: Gibson Hall 126

Time: 3:00

Wednesday, November 28

Algebra and Combinatorics

On the regularity of binomial edge ideals of graphs. Part 2

A.V. Jayanthan - Indian Institute of Technology, Madras

Abstract: TBA

Location: Gibson Hall 325

Time: 3:00

Tuesday, November 27

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Monday, November 26

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Week of November 23 - November 19

Friday, November 23

Thanksgiving Holiday

Thursday, November 22

Thanksgiving Holiday

Wednesday, November 21

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Tuesday, November 20

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Monday, November 19

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Week of November 16 - November 12

Friday, November 16

Applied and Computational

Taming turbulence via nudging

Patricio Clark - University of Rome

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The technique of nudging is commonly used to incorporate empirical data into a simulation in order to control its chaotic evolution and reproduce a given dynamical benchmark. We show how to do this in fully developed three dimensional turbulence using data both in configuration and Fourier space. Our results show that given enough data, nudging is successful in reconstructing the whole turbulent field. We give physical arguments for the choice of optimal parameters and the amount and quality of data needed to do this. Nudging thus serves as a way to probe for key degrees of freedom in a flow. We also turn the algorithm on its head and show how it can be used to infer the values of parameters and the presence of unknown physical mechanisms in the data.

Location: Gibson Hall 310

Time: 3:30

Thursday, November 15

Colloquium

Applications such as electrical-impedance tomography, nanoelectrode sensors, and nanowire sensors lead to deterministic and stochastic partial differential equations that model electrostatics and charge transport. The main model equations are the nonlinear Poisson-Boltzmann equation and the stochastic drift-diffusion-Poisson-Boltzmann system. After a discussion of the model equations, theoretic results as well as a numerical method, namely optimal multi-level Monte Carlo, are presented.

Knowing these model equations, the question how as much information as possible can be extracted from measurements arises next. We use computational Bayesian PDE inversion to reconstruct physical and geometric parameters of the body interior in electrical-impedance tomography and of target molecules in the two nanoscale sensors considered here. Computational Bayesian estimation provides us with the ability not only to estimate unknown parameter values but also their probability distributions and hence the uncertainties in reconstructions, which is important in the case of ill-posed inverse problems. In addition to showing the well-posedness of the Bayesian inversion problem for the nonlinear Poisson-Boltzmann equation, numerical results for the three applications such as multifrequency reconstruction for nanoelectrode sensors are shown.

Location: Location: Gibson Hall 325

Time: 3:30

Thursday, November 15

Geometry and Topology

Density bounds on binary packings of disks in the plane

Ali Mohajer - Tulane University

Abstract:

In this talk we develop methods for establishing upper density bounds for saturated two-radius packings of disks in the plane.

Define the homogeneity of a packing of disks in the plane to be the infimum of the ratio of radii of disks in the packing. It has been known since 1953 (L. Fejes-Toth) that if the homogeneity of a packing is close enough to 1, the density of that packing cannot exceed $\frac{$\pi}{\sqrt{12}},$ the upper bound on the density of a single-radius packing. "Close enough” was refined in 1963 by August Florian to mean a homogeneity in the interval (0.902…, 1], and in 1969, Gerd Blind extended the left bound of this interval to approximately 0.742.

In 2003, sharp upper density bounds were established by Aladar Heppes for a handful of two-radius packings at homogeneities which admit arrangements wherein each disk is tangent to a ring of disks, each of which is tangent to its two cyclic neighbors. We will discuss recent progress in establishing a bound sharper than the best one known for binary packings at a homogeneity that does not admit such regularity.

Location: Gibson Hall 400-D

Time: 12:30

Wednesday, November 14

Probability and Statistics

This talk is about the recovering of stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S\&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Hence, a consistent model for both SPX and VIX derivatives would be one where the SVM is obtained by inverting the market model. A function for stochastic volatility function is the solution of an inverse problem, with the inputs given by a VIX futures market model. Several models are analyzed mathematically and explored numerically.

Location: Gibson Hall 126

Time: 3:00

Wednesday, November 14

Algebra and Combinatorics

Upper bound for the regularity of powers of edge ideals of graphs. Part 1

A.V. Jayanthan - Indian Institute of Technology, Madras

Abstract:

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. In this paper, we obtain an upper bound for reg(I(G)^q) in terms of certain invariants associated with G. We also prove certain weaker versions of a conjecture by Alilooee, Banerjee, Bayerslan and Hà on an upper bound for the regularity of I(G)^q and we prove the conjectured upper bound for the class of vertex decomposable graphs.

Location: Gibson Hall 325

Time: 3:00

Tuesday, November 13

Chudnovsky's Conjecture and Waldschmidt Constant

Sankhaneel Bisui - Tulane University

Abstract:

A well-studied question in algebraic geometry is :

Given a finite set of points in a projective space, what is the minimal degree of a hypersurface that will pass through the points with a given multiplicity? To answer this question Chudnovsky gave a Conjecture using the multiplicity. We are going to see the basic facts of the Conjecture. In this aspect, Waldschmidt constant plays an important role. We will see the general version of the conjecture. Waldschmidt constant has some important connection with LPP. So, if time allows we are going to see some.

Applied and Computational Mathematics

A conditional Gaussian framework for uncertainty quantification, data assimilation and prediction of complex nonlinear turbulent dynamical systems will be introduced in this talk. Despite the conditional Gaussianity, the dynamics remain highly nonlinear and are able to capture strongly non-Gaussian features such as intermittency and extreme events. The conditional Gaussian structure allows efficient and analytically solvable conditional statistics that facilitates the real-time data assimilation and prediction. This talk will include three applications of such conditional Gaussian framework. The first part regards the state estimation and data assimilation of multiscale and turbulent ocean flows using noisy Lagrangian tracers. Rigorous analysis shows that an exponential increase in the number of tracers is required for reducing the uncertainty by a fixed amount. This indicates a practical information barrier. In the second part, an efficient statistically accurate algorithm is developed that is able to solve a rich class of high-dimensional Fokker-Planck equation with strong non-Gaussian features and beat the curse of dimensions. In the last part of this talk, a physics-constrained nonlinear stochastic model is developed, and is applied to predicting the Madden-Julian oscillation indices with strongly non-Gaussian intermittent features. The other related topics such as parameter estimation and causality analysis will also be briefly discussed.

Colloquium

Microorganism locomotion in viscoelastic fluids

Becca Thomases - UC Davis (Host:Fauci, Lisa)

Abstract: Many microorganisms and cells function in complex (non-Newtonian) fluids, which are mixtures of different materials and exhibit both viscous and elastic stresses. For example, mammalian sperm swim through cervical mucus on their journey through the female reproductive tract, and they must penetrate the viscoelastic gel outside the ovum to fertilize. A swimming stroke emerges from the coupled interactions between the complex rheology of the surrounding media and the passive and active body dynamics of the swimmer. We use computational models of swimmers in viscoelastic fluids to understand these interactions. I will show results from several recent investigations, and give mechanistic explanations for some different experimental observations. In particular I will discuss how flexible filaments (such as flagella) can store energy from the fluid to obtain speed enhancements from fluid elasticity.

Location: Gibson Hall 325

Time:3:30

Thursday, November 8

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Thursday, November 8

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Wednesday, November 7

Probability and Statistics

Iterative linear solvers have gained recent popularity due to their computational efficiency and low memory footprint for large-scale linear systems. We will discuss two classical methods, Motzkin's method (MM) and the Randomized Kaczmarz method (RK). MM is an iterative method that projects the current estimation onto the solution hyperplane corresponding to the most violated constraint. Although this leads to an optimal selection strategy for consistent systems, for noisy least squares problems, the strategy presents a tradeoff between convergence rate and solution accuracy. We analyze this method in the presence of noise. RK is a randomized iterative method that projects the current estimation onto the solution hyperplane corresponding to a randomly chosen constraint. We present RK methods which detect and discard corruptions in systems of linear equations, and present probabilistic guarantees that these methods discard all corruptions.

Location:Gibson Hall 126

Time:3:00

Tuesday, November 6

Graduate Student Colloquium

GRAPHS AS REDUCED MODELS FOR DISCRETE FRACTURE NETWORKS

Jaime Lopez - Tulane University

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Discrete fracture networks (DFNs) can be modeled with computationally expensive numerical schemes. We present the formulation of using a graph as a reduced model for a DFN and pose the inversion problem central to this research. We solve the corresponding equations on the graph representation to obtain breakthrough curves, which closely match those created by the high fidelity model. Our solution finds lumped parameters representing the fracture properties, and is used to reduce the computational time required for particle transport calculations. We present examples of creating these reduced models for DFNs with 500 fractures to illustrate the methodology and optimization scheme used to obtain an improved result over a previous formulation.

Location: Stanley Thomas 316

Time: 4:30

Monday, November 5

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Week of November 2 - October 29

Friday, November 2

Special Seminar Applied and Computational

Topic: Mathematics and the Law

Summary
• Criminal law and statistics with real world examples: a serial killer case
• Civil law and statistics with real world examples: a $40,000,000 import case
• Battle of the statisticians from the civil import case
• Q&A with attendees

Dr. Bruce Krell -

Abstract:

The speaker is a former Tulane undergrad with over 43 years experience as an Applied Mathematician. He will show examples of the use of applied statistics in the real world.

Criminal law: The Grim Sleeper was accused of 10 murders based on matching marks from bullets found in the bodies of victims. An examination of the evidence leads to doubts about the underlying science. An example of the use of 3-D laser scans, materials science, hypotheses, and statistics is presented.

Civil law: An importer states the weight of his product. This weight is used to determine import duties. An examination of the sample size and statistical approach by the Customs labs leads to doubts about their methods. An example of the use of sample size calculation, regression, hypothesis testing, and manufacturing quality control is presented.

Location: Gibson Hall 325

Time: 12:00

Friday, November 2

Applied and Computational Mathematics

An Overview of Model Reduction

Christopher Beattie - Virginia Polytechnic Institute and State University

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Dynamical systems form the basic modeling framework for a large variety of complex systems. Direct numerical simulation of these dynamical systems is one of few means available for accurate prediction of the associated physical phenomena. However, ever increasing needs for improved accuracy require the inclusion of ever more detail in the modeling stage, leading inevitably to ever larger-scale, ever more complex dynamical systems that must be simulated. Simulations in such large-scale settings can be overwhelming and may create unmanageably large demands on computational resources; this is the main motivation for model reduction, which has as its goal the extraction simpler dynamical systems that retain essential features of the original systems, especially high fidelity emulation of input/output response and conserved quantities. I will give a brief overview of the objectives and methodology of system theoretic approaches to model reduction, focussing eventually on projection methods that are both simple and capable of providing nearly optimal reduced models in some circumstances. These methods provide a framework for model reduction that allows retention of special model structure such as parametric dependence, passivity/dissipativity, and port-Hamiltonian structure.

Location: Gibson Hall 310

Time: 3:30

Thursday, November 1

Geometry and Topology

Discrete Morse theory and its applications to metric graph reconstruction

Sushovan Majhi - Tulane University

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Discrete Morse theory has recently been used as a powerful tool for topological reconstruction and simplification in the field of Applied and Computational Topology. We discuss some its very recent and successful applications in data analysis. In particular, we talk about a recent development in threshold-based topological and geometric reconstruction of metric graphs from a density concentrated around it.

Location: Gibson Hall 400-D

Time: 12:30

Wednesday, October 31

Probability and Statistics

Universality for Products of Random Matrices

Phil Kopel - University of Colorado Boulder

Abstract:

Random matrices have played an increasingly significant role in diverse areas of pure and applied mathematics over the last fifty years. In the first half of this talk, I will provide a brief overview of the subject: some of the things we study, some of the methods we use, some major results obtained, and several tantalizing unsolved problems. This will hopefully be accessible and informative for a general mathematical audience -- no prior experience required! In the second half of the talk, I will discuss some recent results obtained (in collaboration with Sean O'Rourke and Van Vu) establishing universality of fluctuations in the spectral bulk for products of independent entry ensembles, and outline the proofs.

Location: Gibson Hall 126

Time: 3:00

Wednesday, October 31

Algebra and Combinatorics

We give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a relationship between two seemingly unrelated notions of Commutative Algebra and Combinatorics. It covers all previous major results in this topic and has several interesting consequences.

Location: Gibson Hall 31

Time: 3:00

Tuesday, October 30

Grad Student Colloquium

Normality of Monomial Ideals

Thai Nguyen - Tulane University

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In this talk, I will give an introduction to the concepts of integral closure and normality of rings and ideals and explain why I care about them. Focusing on monomial ideals, I will talk about some approaches to tackle the problem of determining the normality of a monomial ideal provided that some of its ordinary powers are integrally closed. It turns out that there is a surprisingly interesting connection between them and some important objects and problems in convex geometry, graph theory and integer programming.

Location: Stanley Thomas 316

Time:4:30

Monday, October 29

Algebraic Geometry Seminar

Catalan Functions and k-Schur functions

Anna Pun - Drexel University

Abstract:

Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase (n-1, ..., 1,0) (of which there are Catalan many) and a composition weight of length n. They include the Schur functions ,the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture.

A key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are k-Schur functions and proved that graded k-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded k-Schur functions and resolved the Schur positivity and k-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are k-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems.

This is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers.

Location: Gibson Hall 325

Time: 3:00

Week of October 26 - October 22

Friday, October 26

Applied and Computational

A bundled approach for high-dimensional informatics problems

Reginald McGee | College of the Holy Cross

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As biotechnologies for data collection become more efficient and mathematical modeling becomes more ubiquitous in the life sciences, analyzing both high-dimensional experimental measurements and high-dimensional spaces for model parameters is of the utmost importance. We present a perspective inspired by differential geometry that allows for the exploration of complex datasets such as these. In the case of single-cell leukemia data we present a novel statistic for testing differential biomarker correlations across patients and within specific cell phenotypes. A key innovation here is that the statistic is agnostic to the clustering of single cells and can be used in a wide variety of situations. Finally, we consider a case in which the data of interest are parameter sets for a nonlinear model of signal transduction and present an approach for clustering the model dynamics. We motivate how the aforementioned perspective can be used to avoid global bifurcation analysis and consider how parameter sets with distinct dynamic clusters contrast.

Location: Gibson Hall 310

Time: 3:30

Thursday, October 25

Colloquium

In data assimilation that attempts to predict nonlinear evolution of the system by combining complex computational model, observations, and uncertainties associated with them, it is useful to be able to quantify the amount of information provided by an observation or by an observing system. Measures of the observational influence are useful for the understanding of performance of the data assimilation system. The Forecast sensitivity to observation provides practical and useful metric for the assessment of observations. Quite often complex data assimilation systems use a simplified version of the forecast sensitivity formulation based on ensembles. In this talk, we first present the comparison of forecast sensitivity for 4DVar, Hybrid-4DEnVar, and 4DEnKF with or without such simplifications using a highly nonlinear model. We then present the results of ensemble forecast sensitivity to satellite radiance observations for Hybrid-4DEnVart using a global data assimilation system.

Location: Gibson Hall 325

Time: 3:30

Thursday, October 25

AMS/AWM Faculty Talk

From Lindenbaum-Tarski to Stone Duality and beyond

Michael Mislove - Tulane University

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The Lindenbaum-Tarski Theorem states that every complete atomic Boolean algebra is isomorphic to the power set of its set of atoms. It’s easy to turn this into a duality, and the next question is how to generalize to include arbitrary Boolean algebras. This leads to Stone Duality, which has a remarkable number of applications in mathematics and other areas, including in computer science. I’ll outline how some of these results arose, focusing on contributions of two colleagues whose work has influenced my own: Hilary Priestley and Mae Gehrke.

Location: Gibson 400-D

Time: 1:30

Thursday, October 25

Geometry and Topology

Finite-type knot invariants and a proof of the Goussarov Theorem

Robyn Brookr - Tulane University

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Finite-type knot invariants represent an active research area in knot theory. The Goussarov Theorem shows that all such invariants can be read from the Gauss diagram of a knot. In this talk, I will give a proof of this theorem, which provides a method by which one can generate a formula to determine the value of an invariant from its Gauss diagram.

Location: Gibson Hall 400-D

Time: 12:30

Wednesday, October 24

Algebra and Combinatorics

We give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a relationship between two seemingly unrelated notions of Commutative Algebra and Combinatorics. It covers all previous major results in this topic and has several interesting consequences.

Location: Gibson Hall 325

Time: 3:00

Wednesday, October 24

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Tuesday, October 23

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Monday, October 22

Algebraic Geometry

The Variety of Polarizations

Aram Bingham - Tulane University

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Location: Gibson Hall 325

Time: 3:00

Week of October 19 - October 15

Friday, October 19

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Thursday, October 18

Colloquium

Symmetries and choreographies in the N-body problem

Renato Calleja - National Autonomous University of Mexico (Host Glatt-Holtz)

Abstract:

N-body choreographies are periodic solutions to the N-body equations in which N equal masses chase each other around a fixed closed curve. In my talk I will describe numerical and rigorous continuation and bifurcation techniques in a boundary value setting used to follow Lyapunov families of periodic orbits. These arise from the polygonal system of n bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies. I will present a sample of the many choreographies that we have determined numerically along the Lyapunov families and bifurcating families. I will also talk about the computer assisted proofs that validate some of theses choreographies. This is joint work with Eusebius Doedel, Carlos García Azpeitia, Jason Mireles-James and Jean-Philippe Lessard.

Location: Gibson Hall 325

Time: 3:30

Thursday, October 18

AMS/AWM Faculty Student

Academic Job Hunting

Selvi Kara - Institution

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Location: 400-D

Time: 1:30

Thursday, October 18

Geometry and Topology

Topic: On the reconstruction problem for geodesic subspaces in the Euclidean Space

Rafał Komendarczyk - Tulane University

Abstract:

We develop a persistence based algorithm for the homology/homotopy groups reconstruction of the unknown underlying geodesic subspace of R^n from a point cloud. In the case of a metric graph, we can output a subspace which is an arbitrarily good approximation of the underlying graph. This is a collaborative work in progress.

Location: Gibson Hall 400-D

Time:12:30

Wednesday, October 17

Probability and Statistics

Estimating evolutionary rates from pattern in the CRISPR defense memory of prokaryotes

Franz Baumdicker - University of Freiburg

Abstract:

Bacteria and archaea are under constant attack by a myriad of viruses. Consequently, many prokaryotes harbor immune systems against such viral attacks. A prominent example is the CRISPR system, that led to the CRISPR-Cas genome editing technology.

The prokaryotic CRISPR defense system includes an array of spacer sequences that encode an inheritable memory of previous infections. These spacer sequences correspond to short sequence samples from past viral attacks and provide a specific immunity against this particular virus.

Notably, new spacer sequences are always inserted at the beginning of the array, while deletion of spacers can occur at any position in the array. In a sample of CRISPR arrays, spacers can thus be present in all or a subset of the sample, but the order of spacer sequences in the array will be conserved across the sample. This order represents the chronological infection history of the host.
In an evolutionary model for spacer acquisition and deletion we derived the distribution of the number of different spacers between spacers that are present in all arrays. In particular, the order of spacers in the arrays can be used to estimate the rate of spacer deletions independently of the spacer acquisition rate. A property that is usually hard to obtain in population genetics. These estimates provide a basis to study the co-evolution of CRISPR possessing prokaryotes and their viruses.

Location: Gibson Hall 126

Time: 3:00

Tuesday, October 16

Graduate Student Colloquium

Fractional Brownian motion (fBm), whose origins date back to Kolmogorov, is one of the most celebrated models of scale invariance, and has been used in a wide variety of modeling contexts ranging from hydrology to economics. In this talk, I will discuss some background related to scale invariance and fBm, introduce two extensions of fBm: tempered fractional Brownian motion (tfBm), and operator fractional Brownian motion (ofBm), and discuss some recent work related to wavelet-based estimation of tfBm (joint work with G. Didier, F. Sabzikar), as well as some preliminary results regarding estimation ofBm in a high-dimensional setting.

Location: Stanley Thomas 316

Time: 4:30

Monday, October 15

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Week of October 12 - October 8

Friday, October 12

Fall Break

Thursday, October 11

Fall Break

Wednesday, October 10

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Tuesday, October 9

Grad Student Colloquium

Extensions of set partitions

Diego Villamizar - Tulane University

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In this talk we will show arithmetical and combinatorial properties of set partitions that have restrictions in the size of their blocks. This was a joint work with Jhon Caicedo, Victor Moll and Jose Ramirez.

Location: Stanley Thomas 316

Time: 4:30

Monday, October 8

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Week of October 5 - October 1

Friday, October 5

Applied and Computational

How to deduce a physical dynamical model from expectation values

Denys Bondar - Institution

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In this talk, we will provide an answer to the question: "What kind of observations (i.e., expectation values) and assumptions are minimally needed to formulate a physical model?" Our answer to this question leads to the new systematic approach of Operational Dynamical Modeling (ODM), which allows deducing equations of motions from time evolution of observables. Using ODM, we are not only able to re-derive well-known physical theories, but also solve open problems in quantum non-equilibrium statistical dynamics. Furthermore, ODM has revealed unexplored flexibility of nonlinear optics: A shaped laser pulse can drive a quantum system to emit light as if it were a different system (e.g., making lead look like gold).

Location: Gibson Hall 310

Time: 4:10

Thursday, October 4

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Thursday, October 4

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Wednesday, October 3

Algebra and Combinatorics

Constructing a $S_n$ module for $(-1)^{n-1} \nabla p_n$

Speaker - Institution

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We will outline a construction of an $S_n$ module with Frobenius characteristic $(-1)^{n-1} \nabla p_n$. The construction is realized in two steps. First one defines an appropriate sheaf on the Hilbert scheme of points on the plane. Subsequently, one uses Bridgeland-King-Reid correspondence to pass to $S_n$ modules.

Location: Gibson Hall 325

Time: 3:00

Tuesday, October 2

CCS

Algorithms for long- and short-range interactions in soft active matter

Wen Yan - Simons Foundation

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Research Scientist, Biophysical Modeling Group, Center for Computational Biology, Flatiron Institute, Simons Foundation
Algorithms for long- and short-range interactions in soft active matter Soft matter systems often show intriguing phenomena in large spatial scales and long-time scales, due to various long and short-range interactions between the building blocks. The long-range interactions are usually through Stokes flows and Laplace fields, while steric interactions are usually the dominant effect at short-range. The Kernel Independent Fast Multipole Method is extended to various boundary conditions to allow adaptive and flexible treatment of long-range interactions. This algorithm is then extended to a new formulation for half-space Stokes flows induced by point forces or particles. To handle the short-range steric interactions, we propose a new method based on constrained minimization to circumvent the stiffness of pairwise repulsive potential. In this method collision forces are computed based on the geometric constraint that objects do not overlap. All the discussed algorithms are parallel and scalable, and we demonstrate the applications with a few active matter systems, including microtubule network and growing and dividing cells.

Location: Stanley Thomas Hall 316

Time: 3:00

Tuesday, October 2

Grad Students Colloquium

The Mathematics of Music

Nathan Bedell - Institution

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In this talk, I'll explain some of mathematical aspects of music theory. In particular, we will focus on the questions: "Why do we use the 12 notes per octave that we see today on a modern piano keyboard, and not some other collection?" and "Why does most of that music limit itself to certain 5 note (pentatonic) and 7 note (diatonic) subsets of that collection?" respectively.
Our story starts with the history of tuning in the west -- from Pythagorean and meantone temperaments -- including the extended meantone tunings of the Renaissance era, to our modern system of 12 tone equal temperament, some of the various non-western systems of tuning, and finally, to the experimental temperaments that musicians in the "xenharmonic" community have used in recent years to expand the possibilities of our sonic pallet -- all of which will be illustrated with musical examples.

Location: Stanley Thomas 316

Time: 4:30

Monday, October 1

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Week of September 28 - September 24

Friday, September 28

Applied and Computational

Fluid-structure interactions within marine phenomena

Shilpa Khatri - Tulane University

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To understand the fluid dynamics of marine phenomena fluid-structure interaction problems must be solved. Challenges exist in developing analytical and numerical techniques to solve these complex flow problems with boundary conditions at fluid-structure interfaces. I will present details of two different problems where these challenges are handled: (1) modeling of marine aggregates settling in density stratified fluidsand (2)accurate evaluation of layer potentials near boundaries and interfaces. The first problem of modeling marine aggregates will be motivated by field and experimental work. I will discuss the related data and provide comparisons with the modeling. For the second problem of accurate evaluation of layer potentials, I will show how classical numerical methods are problematicfor evaluations close to boundaries and how newly developed asymptotic methods can be used to remove the error. To demonstrate this method, I will consider the interior Laplace problem.

Location: Gibson Hall 310

Time: 3:30

Thursday, September 27

Colloquium

Geometry and topology of random curves

Igor Rivin - Temple University (Host: Victor Moll)

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We study (experimentally and theoretically) random curves (in many senses) in space and in the plane, including such of their properties as the knot type.

Location: Gibson Hall 325

Time: 3:30

Thursday, September 27

AMS/AWM Faculty Talk

Hodge Structures and the Hodge Conjecture

Al Vitter - Tulane University

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The most beautiful and useful invariants used to distinguish between smooth projective varieties with the same topology are the Hodge structures defined on their cohomology groups. I will define Hodge structures and discuss some theorems involving their use. I will also examine the relationship between Hodge structures and sub-varieties, which leads to the Hodge conjecture.

Location: Gibson Hall 400-D

Time: 1:30

Wednesday, September 26

Probability and Statistics

Chase-Escape

Matthew Junge - Duke University

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Imagine barnacles and mussels spreading across the surface of a rock. Barnacles move to adjacent unfilled spots. Mussels too, but they can only attach to barnacles. Barnacles with a mussel on top no longer spread. What conditions on the rock geometry (i.e. graph) and spreading rates (i.e. exponential clocks) ensure that barnacles can survive? Chase-escape can be formalized in terms of competing Richardson growth models; one on top of the other. New, tantalizing open problems will be presented. Joint work with Rick Durrett and Si Tang.

Location: Gibson Hall 126

Time: 3:00

Tuesday, September 25

Graduate Student Colloquium

The Fisher-KPP equation and traveling

Dana Ferranti - Tulane University

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Location: Stanley Thomas 316

Time: 4:30

Monday, September 24

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Week of September 21 - September 17

Friday, September 21

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Thursday, September 20

AMS

Simulations of Pulsating Soft Corals

Shilpa Khatri - UC Merced

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Soft corals of the family Xeniidae have a pulsating motion, a behavior not observed in many other sessile organisms. We are studying how this behavior may give these coral a competitive advantage. We will present direct numerical simulations of the pulsations of the coral and the resulting fluid flow by solving the Navier-Stokes equations coupled with the immersed boundary method. Furthermore, parameter sweeps studying the resulting fluid flow will be discussed.

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Thursday, September 13

Combinatorial structures and invariants through algebraic lenses

Tai Ha - Tulane University

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We shall discuss how to use algebraic formulations to approach and understand a number of problems in linear programming and graph theory. Particularly, we shall examine the (integral) optimal solutions to system of linear constraints, and important invariants in graph theory, such as the chromatic, matching and covering numbers.